It’s the Law — the Laws of Exponents

Summary:
The rules for combining powers and roots seem to confuse a
lot of students. They try to memorize everything, and of course it’s
a big mishmash in their minds. But
the laws just come down to counting, which anyone can do, plus three
definitions to memorize. This page sorts out what you have to memorize
and what you can do based on counting, to solve every problem
involving exponents.

What Is an Exponent, Anyway?

There’s nothing mysterious! An exponent is simply shorthand for
multiplying that number of identical factors.
So 4³ is the same as (4)(4)(4), three identical factors of 4. And
x³ is just three factors of x, (x)(x)(x).

One warning: Remember the order of operations. Exponents are the
first operation (in the absence of grouping symbols like parentheses),
so the exponent applies
only to what it’s directly attached to.
3x³ is 3(x)(x)(x), not (3x)(3x)(3x). If we
wanted (3x)(3x)(3x), we’d need to use grouping: (3x)³.

Negative Exponents

A negative exponent means to divide by
that number of factors instead of multiplying.
So 4−3 is the same as
1/(43), and x-3 = 1/x3.

As you know, you can’t divide by zero. So there’s a
restriction that x−n = 1/xn only when x is
not zero. When x = 0, x−n is undefined.

Fractional Exponents

A fractional exponent—specifically, an exponent of the
form 1/n—means to take the nth root
instead of multiplying or dividing. For example, 4(1/3) is
the 3rd root (cube root) of 4.

Arbitrary Exponents

You can’t use counting techniques on an expression like
60.1687 or 4.3π. Instead, these
expressions are
evaluated using logarithms.

Here’s All You Need to Memorize

And that’s it for memory work. Period. If you memorize
these three definitions, you can work everything else out by combining
them and by counting:

Granted, there’s a little bit of hand waving in my statement that
you can work everything else out. Let me make good on that promise, by
showing you how all the other laws of exponents come from just the three
definitions above. The idea is that you won’t need to memorize the
other laws—or if you do choose to memorize them,
you’ll know why they work and you’ll find them easier to memorize
accurately.

Multiplying and Dividing Powers

Two Powers of the Same Base

Suppose you have (x5)(x6); how do you
simplify that? Just remember that you’re counting factors.

x5 = (x)(x)(x)(x)(x)
and x6 = (x)(x)(x)(x)(x)(x)

Now multiply them together:

(x5)(x6) =
(x)(x)(x)(x)(x)(x)(x)(x)(x)(x)(x) = x11

Why x11? Well, how many x’s are there? Five x
factors from x5, and six x factors from x6,
makes 11 x factors total. Can you see that whenever you multiply
any two powers of the same base, you end up with a number of
factors equal to the total of the two powers? In other words,
when the bases are the same, you find the
new power by just adding the exponents:

Powers of Different Bases

Caution! The rule above works only when multiplying powers
of the same base. For instance,

(x3)(y4) =
(x)(x)(x)(y)(y)(y)(y)

If you write out the powers, you see there’s no
way you can combine them.

Except in one case:
If the bases are different but the exponents are the same,
then you can combine them. Example:

(x³)(y³) = (x)(x)(x)(y)(y)(y)

But you know that it doesn’t matter what order
you do your multiplications in or how you group them. Therefore,

(x)(x)(x)(y)(y)(y) =
(x)(y)(x)(y)(x)(y) = (xy)(xy)(xy)

But
from the very definition of powers, you know that’s the same as
(xy)³. And it works for any common power of two different
bases:

It should go without saying, but I’ll say it
anyway: all the laws of exponents work in both directions. If you see
(4x)³ you can decompose it to (4³)(x³), and if you see
(4³)(x³) you can combine it as (4x)³.

Dividing Powers

What about dividing? Remember that dividing is just multiplying
by 1-over-something. So all the laws of division are really just laws
of multiplication. The extra definition of x-n as
1/xn comes into play here.

Example: What is x8÷x6? Well, there are
several ways to work it out. One way is to say that
x8÷x6 =
x8(1/x6), but using the definition of
negative
exponents that’s just x8(x-6). Now use the
product rule (two powers of the same
base) to rewrite it as
x8+(-6), or x8-6, or x2. Another
method is simply to go back to the definition:
x8÷x6 =
(xxxxxxxx)÷(xxxxxx) = (xx)(xxxxxx)÷(xxxxxx) =
(xx)(xxxxxx÷xxxxxx) = (xx)(1) = x2. However
you slice it, you come to the same answer:
for division with like bases you subtract exponents,
just as for multiplication of like
bases you add exponents:

But there’s no need to memorize
a special rule for division: you can always work it out from the other
rules or by counting.

In the same way, dividing different bases
can’t be simplified unless the exponents are equal.
x³÷y² can’t be combined because it’s just xxx/yy; But
x³÷y³ is xxx/(yyy), which is (x/y)(x/y)(x/y), which is
(x/y)³.

Multiplication and division have equal
precedence, so xxx/yyy would literally mean x, times x, times x,
divided by y, times y, times y and would be equal to xxxy.
That’s why the parentheses are necessary: xxx/(yyy), as reader
Chase Ries pointed out. I had written xxx/yyy, because we often omit
the parentheses in a fraction that
doesn’t contain additions or
subtractions. But it’s best not to force the reader to puzzle out
from the context whether some parentheses have been omitted.

Parentheses around the xxx —
(xxx)/(yyy) — would not be wrong, but they’re
not needed because the standard order of operations is to
multiply x by x by x, with or without parentheses.

As that example illustrates, you can combine like exponents
even when the bases are different:

Negative Powers on the Bottom

What about dividing by a negative power, like
y5/x−4? Use the rule you already know for
dividing:

How many factors of x are there? You see that
there are 5 factors in each row from x5 and 4 rows from
( )4, in all 5×4=20 factors. Therefore,

(x5)4 = x20

As you might expect, this applies to any
power of a power: you multiply the exponents. For
instance, (k-3)-2 =
k(-3)(-2) = k6. In general,

I can just hear you asking, “So when do I add exponents
and when do I multiply exponents?” Don’t try to remember a
rule—work it out! When you have a
power of a power, you’ll always have a
rectangular array of factors, like the example above. Remember the old
rule of length×width, so the combined exponent is formed by
multiplying. On the other hand, when you’re only multiplying two powers together, like
g2g3, that’s just the same as stringing factors
together,

g2g3 = (gg)(ggg) =
(ggggg) = g5

You can always refresh your memory by counting
simple cases, like

x2x3 = (xx)(xxx) =
x5

versus

(x2)3 = (xx)3 = (xx)(xx)(xx) =
x6

Now You Try It!

The Zero Exponent

You probably know that anything to the 0 power
is 1. But now you can see why. Consider x0.

By the division rule, you know that
x3/x3 = x(3−3) = x0.
But anything divided by itself is 1, so
x3/x3 = 1. Things that are equal to
the same thing are equal to each other: if x3/x3
is equal to both 1 and x0, then 1 must equal x0.
Symbolically,

x0 = x(3−3) =
x3/x3 = 1

There’s one restriction. You saw that we had to
create a fraction to figure out x0. But division by 0 is
not allowed, so our evaluation works for anything to the 0 power
except zero itself:

Evaluating 00 is a topic for your calculus course.

Now You Try It!

Radicals

The laws of radicals are traditionally taught separately from the
laws of exponents, and frankly I’ve never understood why. A
radical is simply a fractional exponent:
the square (2nd) root of x is just x1/2, the cube (3rd)
root is just x1/3, and so on. With this fact at your
disposal, you’re in good shape.

Example: . That’s easy to evaluate! You know that the cube
(3rd) root of x is x1/3 and the square root of that is
(x1/3)1/2. Then use the
power-of-a-power rule to evaluate that as
x(1/3)(1/2) = x(1/6), which is the 6th root of x.

Fractional or Rational Exponents

So far we’ve looked at fractional exponents only where the top
number was 1. How do you interpret x2/3, for instance?
Can you see how to use the power rule? Since
2/3 = (2)(1/3), you can rewrite x2/3 =
x(2)(1/3) = (x2)1/3, which is
. It works the other way, too: 2/3 =
(1/3)(2), so x2/3 =
x(1/3)(2) =
(x1/3)2 =
. These are examples of the general rule:

When a power and a root are involved, the top part of
the fractional exponent is the power and the bottom part is the
root.

Suppose p and r are the same? Then you have, for instance,
. But that’s the same as x5/5, and
5/5=1, so it’s the same as x1 or just x.

Now You Try It!

Conclusion

Well, there you are: the laws of exponents and radicals
demystified! Just remember the
three basic definitions. When you’re not sure
about a rule, like the product rule, don’t try to remember it, just
work it out by counting and you’ll do just fine.